Lesson plan
Zeta function (operator)
The zeta function of a mathematical operator \mathcal O is a function defined as : \zeta_{\mathcal O}(s) = \operatorname{tr} \; \mathcal O^{-s} for those values of s where this expression exists, and as an analytic continuation of this function for other values of s. Here "tr" denotes a functional trace. The zeta function may also be expressible as a spectral zeta function in terms of the eigenvalues \lambda_i of the operator \mathcal O by : \zeta_{\mathcal O}(s) = \sum_{i} \lambda_i^{-s} . It is used in giving a rigorous definition to the functional determinant of an operator, which is given by : \det \mathcal O := e^{-\zeta'_{\mathcal O}(0)} \;. The Minakshisundaram–Pleijel zeta function is an example, when the operator is the Laplacian of a compact Riemannian manifold. One of the most important motivations for Arakelov theory is the zeta functions for operators with the method of heat kernels generalized algebro-geometrically.
Article summary
The zeta function of a mathematical operator \mathcal O is a function defined as : \zeta_{\mathcal O}(s) = \operatorname{tr} \; \mathcal O^{-s} for those values of s where this expression exists, and as an analytic continuation of this function for other values of s. Here "tr" denotes a functional trace. The zeta function may also be expressible as a spectral zeta function in terms of the eigenvalues \lambda_i of the operator \mathcal O by : \zeta_{\mathcal O}(s) = \sum_{i} \lambda_i^{-s} . It is used in giving a rigorous definition to the functional determinant of an operator, which is given by : \det \mathcal O := e^{-\zeta'_{\mathcal O}(0)} \;. The Minakshisundaram–Pleijel zeta function is an example, when the operator is the Laplacian of a compact Riemannian manifold. One of the most important motivations for Arakelov theory is the zeta functions for operators with the method of heat kernels generalized algebro-geometrically.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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